Why can't cycles cast one ray per image pixel, instead of rendering progressively to infinity?
That is exactly what happens if you set the numer of samples to 1. You can try doing that. The result you get, however, will be unsatisfying.

The problem is that every sample can only travel one path each time it hits a surface. But most surfaces allow a lot more than one reflection direction. A diffuse surface is characterized, for example, by reflecting incoming light in an almost completely random direction.

So if you sample Pixel 1, and the light hits the first surface and happens to reflect, say, towards the dark night sky, this ray will tell the renderer that it shot to ~infinity and found no light. So this pixel is black.
On the pixel next to it, the ray, due to the randomness of the reflection of the surface, will happen to be reflected to the right, towards a big lamp, and will find a lot of light traveling this ray path, so this pixel will be very bright.

If you repeat this for all pixels, the result for each pixel is very random and the image you get is very, very noisy (more so the smaller your light sources are).

So, essentially, the reason why you need to shoot several times per pixel is because you have to shoot enough times that for every pixel, most directions have been "probed" and they all give approximately the same result.

If you only render perfect mirrors with no variation in reflection direction, one sample per pixel is enough, nothing will change with the second one. This is very rare as a scene though.